Flows of $G_2$-Structures associated to Calabi-Yau Manifolds

TL;DR

This paper links the $G_2$-Laplacian flow to complex Monge-Ampère equations on Calabi-Yau manifolds, proving long-time existence and convergence to torsion-free structures derived from Ricci-flat Kähler metrics.

Contribution

It establishes a novel correspondence between $G_2$-flows and complex Monge-Ampère equations, demonstrating long-term behavior and convergence for specific initial data.

Findings

Flow exists for all time for certain initial data

Flow converges to torsion-free $G_2$-structures

Related coflow aligns with Kähler-Ricci flow

Abstract

We establish a correspondence between a parabolic complex Monge-Amp\`ere equation and the $G_2$-Laplacian flow for initial data produced from a K\"ahler metric on a complex $2$- or $3$-fold. By applying estimate for the complex Monge-Amp\`ere equation, we show that for this class of initial data the $G_2$-Laplacian flow exists for all time and converges to a torsion-free $G_2$-structure induced by a K\"ahler Ricci-flat metric. Similar results are obtained for the $G_2$-Laplacian coflow, and in this case the coflow is related to the K\"ahler-Ricci flow.